Algebra - I

Foundational PG level course in algebra, suitable for M.Sc and first year PhD students in Mathematics.INTENDED AUDIENCE :Any Interested LearnersPREREQUISITES :BSc-level linear algebraINDUSTRIES SUPPORT :Not Applicable

Week 1:Permutation Groups, symmetries of graphsWeek 2:Abstract Groups, homomorphism, isomorphism, subgroup, quotient groupWeek 3:Group actionsWeek 4:Sylow theorems
Week 5:Examples and definition of ringsWeek 6:Homomorphisms, ideals, quotients, integral domains, fraction fieldsWeek 7:Principal ideal domains, Unique factorization domains and Euclidean domainsWeek 8:Modules, definition and examples, submodules, simplicity, indecomposability
Week 9:Tensor productsWeek 10:Jordan-Holder and Krull-Schmidt theoremsWeek 11:Classification of finitely generated modules over PID’sWeek 12:Applications of classification to finitely generated abelian groups and Jordan canonical form